Game theory and the penalty shootout

The week before last in ECON100 we covered game theory. New Zealand is currently hosting the FIFA U20 World Cup, and Christoph Schumacher and Nigel Espie from Massey University have been writing a series of articles in the New Zealand Herald, linking the World Cup to business. Last week they wrote about penalty shootouts:

Seen through a game theory lens, a penalty shootout is a game that involves two players - the striker and the keeper. To simplify matters, let's assume that both players have three options. The striker can choose to make their shot to the right, centre, or left of the goal. Similarly, the keeper can elect to dive to the left or right, or defend the centre of the goal.

Let's also assume the striker is equally good at taking shots to all three areas in the goal and the keeper is equally good at saving balls kicked to the three sections of the goal. So what would be the best strategy for a goalkeeper in this situation?

The penalty shootout is an excellent example of a game with no Nash equilibrium in pure strategy - where the equilibrium is a mixed strategy equilibrium. That is, it is best for the players to randomise their strategy choice. As Schumacher and Espie put it:

With no past knowledge of the striker's penalty history, you would expect the striker to randomly select their shot. Game theory would suggest that the best option for the keeper is to also randomly select an area of the goal to defend. This maximises the likelihood of saving a goal while preventing the opposing team from discerning any preference by the keeper.

To see why, let's set up the game as we would in ECON100. It's a simultaneous game, because even though the shooter makes their choice first, it's unlikely that the goalkeeper has enough time to properly observe the shooter's choice before choosing their own strategy. So, we can represent the game in a payoff table (see below). The shooter has three strategy options (left, centre, right), and the goalkeeper has three strategy options (left, centre, right). To keep things simple, let's assume that if both players choose the same strategy then the goalkeeper saves the shot (the goalkeeper receives a payoff of +1, and the shooter receives a payoff of -1), and if both players choose different strategies, the goal is scored (the goalkeeper receives a payoff of -1, and the shooter receives a payoff of +1). The payoff tables looks as follows:

To confirm that there are no pure strategy Nash equilibriums, we can use the 'best response' method. To do this, we track: for each player, for each strategy, what is the best response of the other player. Where both players are selecting a best response, they are doing the best they can, given the choice of the other player (this is the definition of Nash equilibrium).

For our game outlined above:

1. If the shooter shoots to the left, the goalkeeper's best response is to dive to the left (since +1 is better than -1) [we track the best responses with ticks, and not-best-responses with crosses; Note: I'm also tracking which payoffs I am comparing with numbers corresponding to the numbers in this list];
2. If the shooter shoots to the centre, the goalkeeper's best response is to remain in the centre (since +1 is better than -1);
3. If the shooter shoots to the right, the goalkeeper's best response is to dive to the right (since +1 is better than -1);
4. If the goalkeeper dives to the left, the shooter's best response is to shoot to the centre or to the right (since +1 is better than -1);
5. If the goalkeeper remains in the centre, the shooter's best response is to shoot to the left or to the right (since +1 is better than -1); and
6. If the goalkeeper dives to the right, the shooter's best response is to shoot to the left or to the centre (since +1 is better than -1).

Notice that there are no outcomes (cells) where both players are playing a best response, so there is no Nash equilibrium in pure strategy. To work out the mixed strategy equilibrium, we would need to know a bit more about the probability of success if both goalkeeper and shooter chose the same (here we assume there was a 0% chance of a goal), and the probability of success if they chose differently (here we assumed there was a 100% chance of a goal). Moreover, these probabilities might be different depending on whether it was to the left or right. The equilibrium would therefore not necessarily be random and symmetric.

Sports are filled with mixed strategy equilibriums. Think about tennis serving (down the centre, into the body, out wide), or American football offense (pass vs. run), to name just two. In these (and many other cases), it is best to randomise your actions, because if you become too predictable, your rival can take advantage of that.